This study focuses on two new aspects of inverse modelling of volcanic emissions. First, we derive an observation operator for satellite retrievals of plume height, and second, we solve the inverse problem using an algorithm based on the 4D-Var data assimilation method. The approach is first tested in a twin experiment with simulated observations and further evaluated by assimilating IASI SO₂ plume height and total column retrievals in a source term inversion for the 2010 eruption of Eyjafjallajökull. The inversion resulted in temporal and vertical reconstruction of the SO₂ emissions during 1–20 May 2010 with formal vertical and temporal resolutions of 500 m and 12 h.

The plume height observation operator is based on simultaneous assimilation of the plume height and total column retrievals. The plume height is taken to represent the vertical centre of mass, which is transformed into the first moment of mass (centre of mass times total mass). This makes the observation operator linear and simple to implement. The necessary modifications to the observation error covariance matrix are derived.

Regularization by truncated iteration is investigated as a simple and efficient regularization method for the 4D-Var-based inversion. In the twin experiments, the truncated iteration was found to perform similarly to the commonly used Tikhonov regularization, which in turn is equivalent to a Gaussian a priori source term. However, the truncated iteration allows the level of regularization to be determined a posteriori without repeating the inversion.

In the twin experiments, assimilating the plume height retrievals resulted in a 5–20 % improvement in root mean squared error but simultaneously introduced a 10–20 % low bias on the total emission depending on assumed emission profile. The results are consistent with those obtained with real data. For Eyjafjallajökull, comparisons with observations showed that assimilating the plume height retrievals reduced the overestimation of injection height during individual periods of 1–3 days, but for most of the simulated 20 days, the injection height was constrained by meteorological conditions, and assimilation of the plume height retrievals had only small impact. The a posteriori source term for Eyjafjallajökull consisted of 0.29 Tg (with total column and plume height retrievals) or 0.33 Tg (with total column retrievals only) erupted SO₂ of which 95 % was injected below 11 or 12 km, respectively.